Simplified gentlest ascent dynamics for saddle points in non-gradient systems.

Area of Science:

• Dynamical Systems Theory
• Computational Physics
• Nonlinear Dynamics

Background:

• Saddle points are crucial for understanding activated processes in randomly perturbed dynamical systems.
• The original gentlest ascent dynamics (GAD) efficiently locates saddle points but requires multiple direction variables for non-gradient systems.
• Approximating eigenvectors of Jacobian and transposed Jacobian matrices is computationally intensive.

Purpose of the Study:

• To present a simplified gentlest ascent dynamics (GAD) method requiring only one direction variable for non-gradient systems.
• To reduce computational cost and eliminate the need for Jacobian matrix transpose operations.
• To demonstrate the efficacy of the simplified GAD in various non-gradient systems.

Main Methods:

• Development of a simplified GAD algorithm using a single direction variable.
• Mathematical proof of the convergence property of the simplified GAD.
• Application of the simplified GAD to non-gradient examples, including a 2D model and the Allen-Cahn equation with shear flow.

Main Results:

• The simplified GAD successfully locates saddle points in non-gradient systems using only one direction variable.
• Computational cost for direction variable calculation is halved compared to the original GAD.
• The method avoids the computationally expensive transpose operation of the Jacobian matrix.
• Convergence properties of the simplified GAD are proven to be equivalent to the original GAD.

Conclusions:

• The simplified GAD offers a more computationally efficient and practical approach for locating saddle points in non-gradient systems.
• This method retains the accuracy and convergence of the original GAD while reducing resource requirements.
• The simplified GAD is a valuable tool for studying activated processes and noise-induced dynamics.